#369046

order	512
nilpotency class	3
isoclinism	isoclinic class of #196152
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1842 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(2,3)(5,6)(7,11)(8,18,21,33)(9,13)(10,14)(12,20,27,34)(15,30,32,36)(16,24)(17,25)(19,29)(22,28)(23,31,26,35);(1,2,4,3)(5,7,10,17)(6,11,14,25)(8,15)(9,22,19,16)(12,26)(13,28,29,24)(18,30)(20,35)(21,32)(23,27)(31,34)(33,36);(5,8,19,34,10,21,9,20)(6,12,13,18,14,27,29,33)(7,15,16,31,17,32,22,35)(11,23,28,36,25,26,24,30);(1,3,4,2)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=fg,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276242);